where \(f:\mathbb {R}^d\rightarrow \mathbb {R}\)  is a stochastic objective function that involves a random variable \(\xi \sim {\mathcal {P}}\) , following the distribution \({\mathcal {P}}\) , and \(c:\mathbb {R}^d\rightarrow \mathbb {R}^m\)  provides deterministic equality constraints.
Problems of this form appear widely in a variety of applications, including optimal control [1]}, multi-stage optimization [2]}, PDE-constrained optimization [3]}, constrained maximum likelihood estimation [4]}, and constrained deep neural networks [5]}.
In practice, the variable \(\xi \)  corresponds to a data point; \(f({x}; \xi )\)  is the loss at data point \(\xi \)  when using parameter \({x}\)  to fit the model; and \(f({x})\)  is the expected loss.
Deterministic constraints are also common in many real examples. They can characterize the physics of systems, encode prior model knowledge, or reduce searching complexity.
